Thomas LorenzMutational AnalysisA Joint Framework for Cauchy ProblemsIn and Beyond Vector SpacesPrefaceDifferential problems should not be restricted to vector spaces in general.The main goal of this bookOrdinary differential equations play a central role in science. Newton’s SecondLaw of Motion relating force, mass and acceleration is a very famous and oldexample formulated via derivatives. The theory of ordinary differential equationswas extended from the ﬁnite-dimensional Euclidean space to (possibly inﬁnite-dimensional) Banach spaces in the course of the twentieth century. These so-calledevolution equations are based on strongly continuous semigroups.For many applications, however, it is difﬁcult to specify a suitable normed vectorspace. Shapes, for example, do not have an obvious linear structure if we dispensewith any a priori assumptions about regularity and thus, we would like to describethem merely as compact subsets of the Euclidean space.Hence, this book generalizes the classical theory of ordinary differential equationsbeyond the borders of vector spaces, which is just a tradition from our point of view.It focuses on the well-posed Cauchy problem in any ﬁnite time interval.In other words, states are evolving in a set (not necessarily a vector space) and, theydetermine their own evolution according to a given “rule” concerning their current“rate of change” — a form of feedback (possibly even with ﬁnite delay).

Thomas LorenzMutational AnalysisA Joint Framework for Cauchy ProblemsIn and Beyond Vector SpacesPrefaceDifferential problems should not be restricted to vector spaces in general.The main goal of this bookOrdinary differential equations play a central role in science. Newton’s SecondLaw of Motion relating force, mass and acceleration is a very famous and oldexample formulated via derivatives. The theory of ordinary differential equationswas extended from the ﬁnite-dimensional Euclidean space to (possibly inﬁnite-dimensional) Banach spaces in the course of the twentieth century. These so-calledevolution equations are based on strongly continuous semigroups.For many applications, however, it is difﬁcult to specify a suitable normed vectorspace. Shapes, for example, do not have an obvious linear structure if we dispensewith any a priori assumptions about regularity and thus, we would like to describethem merely as compact subsets of the Euclidean space.Hence, this book generalizes the classical theory of ordinary differential equationsbeyond the borders of vector spaces, which is just a tradition from our point of view.It focuses on the well-posed Cauchy problem in any ﬁnite time interval.In other words, states are evolving in a set (not necessarily a vector space) and, theydetermine their own evolution according to a given “rule” concerning their current“rate of change” — a form of feedback (possibly even with ﬁnite delay). In parti-cular, the examples here do not have to be gradient systems in metric spaces.The driving force of generalization: Solutions via Euler methodThe step-by-step extension starts in metric spaces and ends up in nonempty setsthat are merely supplied with suitable families of distance functions (not necessar-ily symmetric or satisfying the triangle inequality).Solutions to the abstract Cauchy problem are usually constructed by means of Eu-ler method and so, the key question for each step of conceptual generalization is:Which aspect of the a priori given structures can be still weakened so that Eulermethod does not fail ?Diverse examples have always given directions ... towards a joint framework.In the 1990s, Jean-Pierre Aubin suggested what he called mutational equations andapplied them to systems of ordinary differential equations and time-dependent com-Npact subsets of R (supplied with the popular Pompeiu-Hausdorff metric). They arethe starting point of this monograph.Further examples, however, reveal that Aubin’s a priori assumptions (about the addi-tional structure of the metric space) are quite restrictive indeed. There is no obviousway for applying the original theory to semilinear evolution equations.viOur basic strategy to generalize mutational equations is simple: Consider severaldiverse examples successively and, whenever it does not ﬁt in the respective muta-tional framework, then ﬁnd some extension for overcoming this obstacle.Mutational Analysis is deﬁnitely not just to establish another abstract term of solu-tion though. Hence, it is an important step to check for each example individuallywhether there are relations to some more popular meaning (like classical, strong,weak or mild solution).Here are some of the examples under consideration in this book:N– Feedback evolutions of nonempty compact subsets of R– Semilinear evolution equations in arbitrary Banach spaces– Nonlocal parabolic differential equations in noncylindrical domainsN– Nonlinear transport equations for Radon measures on R+– Structured population model with Radon on R0– Stochastic ordinary differential equations with nonlocal sample dependence... and these examples can now be coupled in systems immediately – due to thejoint framework of Mutational Analysis. This possibility provides new tools formodelling in future.The structure of this extended book ... for the sake of the readerThis monograph is written as a synthesis of two aims: ﬁrst, the reader should havequick access to the results of individual interest and second, all mathematical con-clusions are presented in detail so that they are sufﬁciently comprehensible.Each chapter is elaborated in a quite self-contained way so that the reader has theopportunity to select freely according to the examples of personal interest. Hencesome arguments typical for mutational analysis might make a frequently repeatedimpression, but they are always adapted to the respective framework. Moreover, theproofs are usually collected at the end of each subsection so that they can be skippedeasily if wanted. References to results elsewhere in the monograph are usually sup-plied with page numbers. Each example contains a table that summarizes the choiceof basic sets, distances etc. and indicates where to ﬁnd the main results.Introductory Chapter 0 summarizes the essential notions and motivates the gen-eralizations in this book. Many of the subsequent conclusions have their origins in§§ 1.1 – 1.6 and so, these subsections facilitate understanding the modiﬁcations later.Experience has already taught that such a monograph cannot be written free fromany errors or mistakes. I would like to apologize in advance and hope that the gistof both the approach and examples is clear. Comments are cordially welcome.Heidelberg, winter 2009 Thomas LorenzviiAcknowledgmentsThis monograph would not have been elaborated if I had not beneﬁted from the har-mony and the support in my vicinity. Both the scientiﬁc and the private aspect areclosely related in this context.Prof. Willi Jager¨ has been my academic teacher since my very ﬁrst semesterat Heidelberg University. Infected by the “virus” of analysis, I have followed hiscourses for gaining insight into mathematical relations. As a part of his scientiﬁcsupport, he drew my attention to set-valued maps quite early and gave me the op-portunity to gain experience very autonomously. Hence I would like to express mydeep gratitude to Prof. Jager¨ .Moreover, I am deeply indebted to Prof. Jean–Pierre Aubin and Hel´ ene` Frankowska.Their mathematical inﬂuence on me started quite early — as a consequence of theirmonographs. During three stays at CREA of Ecole Polytechnique in Paris, I bene-ﬁted from collaborating with them and meeting several colleagues sharing my math-ematical interests partly.Furthermore, I would like to thank all my friends, collaborators and colleagues re-spectively for the inspiring discussions and observations over time. This list (in al-phabetical order) is neither complete nor a representative sample, of course: ZviArtstein, Robert Baier, Bruno Becker, Hans Belzer, Christel Bruschk¨ e, Eva Cruck,¨Roland Dinkel, Herbert-Werner Diskut, Tzanko Donchev, Matthias Gerdts, PiotrGwiazda, Peter E. Kloeden, Roger Kompf,¨ Stephan Luckhaus, Anna Marciniak-´ ´Czochra, Reinhard Mohr, Jerzy Motyl, Jose Alberto Murillo Hernandez, JanoschRieger, Ina Scheid, Ursula Schmitt, Roland Schnaubelt, Oliver Schnurer¨ , JensStarke, Angela Stevens, Martha Stocker, Manfred Taufertshofer¨ , Friedrich Tomi,Edelgard Weiß-Bohme,¨ Kurt Wolber.Heidelberg University and, in particular, its Interdisciplinary Center for ScientiﬁcComputing (IWR) has been my extraordinary home institutions so far. In addition,some results of this monograph were elaborated as parts of projects or during re-search stays funded by– German Research Foundation DFG (SFB 359 and LO 273)– Hausdorff Institute for Mathematics in Bonn (spring 2008)– Research Training Network “Evolution Equations for Deterministic andStochastic Systems” (HPRN–CT–2002–00281) of the European Community– Minerva Foundation for scientiﬁc cooperation between Germany and Israel.Finally, I would like to express my deep gratitude to my family.My parents have always supported me and have provided the harmonic vicinity sothat I have been able to concentrate on my studies. Surely I would not have reachedmy current situation without them as a permanent pillar.Meanwhile my wife Irina Surovtsova is at my side for several years. I have alwaystrusted her to give me good advice and so, she has often enabled me to overcomeobstacles — both in everyday life and in science. I am optimistic that together wecan cope with the challenges that Daniel, Michael and the “other aspects” of lifeprovide for us. TLviiiContentsPreface ........................................................ vAcknowledgments .............................................. vii0 Introduction ................................................... 10.1 Diverse evolutions come together under the same roof . ........... 10.2 Extending the traditional horizon: Evolution equationsbeyond vector spaces ....................................... 30.2.1 Aubin’s initial notion: Regard afﬁne linear maps just asa special type of “elementary deformations”. . ............ 30.2.2 Mutational analysis as an “adaptive black box”for initial value problems.............................. 60.2.3 The initial problem decomposition and the ﬁnal linkto more popular meanings of abstract solutions ........... 80.2.4 The new steps of generalization ........................ 90.3 Mutational inclusions ....................................... 201 Extending ordinary differential equations to metric spaces:Aubin’s suggestion ............................................. 211.1 The key for avoiding (afﬁne) linear structures: Transitions . . ...... 211.2 The mutation as counterpart of time derivative . . ................ 271.3 Feedback leads to mutational equations ........................ 281.4 Proofs for existence and uniqueness of solutionswithout state constraints . .................................... 301.5 An essential advantage of mutational equations:Solutions to systems ........................................ 341.6 Proof for existence of solutions under state constraints ........... 371.7 Some elementary properties of the contingent transition set . ...... 41N1.8 Example: Ordinary differential equations in R ................. 43ixx ContentsN1.9 Example: Morphological equations for compact sets in R ........ 471.9.1 The Pompeiu-Hausdorff distance dl ..................... 47N1.9.2 transitions on (K (R ), dl ) ............... 501.9.3 Morphological primitives as reachable sets 541.9.4 Some examples of morphological primitives.............. 561.9.5 Some e of contingent transition sets 571.9.6 Solutions to morphological equations ................... 641.10 Example: Modiﬁed equationsvia bounded one-sided Lipschitz maps ......................... 692 Adapting mutational equations to examples in vector spaces ........ 752.1 The topological environment of this chapter .................... 76